ECIV 301

Programming & Graphics

Numerical Methods for Engineers

Lecture 10

Roots of Polynomials

Objective

nnn xaxaxaaxf 2

210

Calculate Roots of a polynomial

• There are n complex or real roots

• If n odd, at least one real root

• Complex roots exist in conjugate pairs

Polynomial Evaluation

33

2210 xaxaxaaxfn

xaaxxaaxfn 322

10

xaaxaxaxfn 3210

Nested Form

Polynomial Evaluation

33

2210 xaxaxaaxfn

xaaxaxaxfn 3210

6 Multiplications 4 additions

3 Multiplications 3 additions

Polynomial EvaluationIn general

nnn xaxaxaaxf 2

210

xaaxaxaxf nnn 110

n(n+1)/2 Multiplications n Additions

n Multiplications n Additions

Evaluating the Derivative

xaaxaxaxfn 3210

dx

xaaxadx

xaaxadx

xd

dx

xdfn

321

321

dx

xaadxxaa

dx

xdx

xaaxadx

xd

dx

xdfn

3232

321

332

321

adx

xdxxaa

dx

xdx

xaaxadx

xd

dx

xdfn

Algorithm

xaaxaxaxf nnn 110

nn aaaaa 110 Store Coefficients in Vector Form

Evaluate starting from Most Inner Parenthesis

xaaxf nnnn 1

Algorithm

xaaxaxaxf nnn 110

Continue by adding terms

xxfaxf nnnn

21

xxfaxfn 212

xxfaxfn 101

Pseudo Codep=0df=0

DO j=n,0,-1

df = df * x + p

p = p * x + a(j)

ENDDO

Must be evaluated First

Polynomial Deflation

54325 xx7x3x79x46120xf

2x3x5x4x1xxf5

Can also be expressed in factored form

Polynomial Deflation

2x5x4x1xxf4

Divide by any of the factors, e.g. (x+3)

432 xx10x27x240

2x3x5x4x1xxf5

Deflation – Division by monomial (x-t)

tx/xfn

nn aaaaa 110 Store Coefficients in Vector Form

Deflation – Division by monomial (x-t)

nar

0a n Do i=n-1, 0, -1

iasra i

t*rsr EndDo

Polynomial Division

Muller’s Method

Muller’s - Algorithm

Define Initial Guesses xo, x1, x2

01o xxh

121 xxh

(1)

(2)

For Each Iteration

Muller’s - Algorithm

0

01o h

xfxf

1

121 h

xfxf

(3)

(4)

Muller’s - Algorithm

01

01

hha

(5)

11ahb (6) 11ahb

(7) 2xfc

Muller’s - Algorithm

(8)

ac4bb

c2xx

223

Compute New Estimate

Two Roots: Choose sign that agrees with sign of b

Muller’s - Algorithm

%100x

xx

3

23a

(9) Compute Error

If converged FINISH

Muller’s - Algorithm

(10) Assign new guesses

(A) For Real Roots ONLYChoose the two points that are closest to x3

(B) For ALL Roots

322110 xxxxxx

NEXT ITERATION(11)